1. Field of the Invention
The present invention relates to a semiconductor device simulation method, in particular, to a simulation method which evaluates the device characteristics of a MOSFET (Metal-Oxide Semiconductor Field-Effect-Transistor) with a higher degree of accuracy and at a higher speed by a numerical simulation.
2. Description of Related Art
In a simulation, distributions of electric potential, electron density, and hole density inside the device when a voltage is applied to the electrodes under a preset device configuration or impurity distribution are obtained numerically. Using these numerical values, the current, the distribution of the electric field, and the electrode current inside the device are obtained. In general, a system of simultaneous equations consisting of the following Poisson equation, equation of continuity of electrons, equation of continuity of holes, electron transport equation, and hole transport equation is solved for electric potential "psgr", electron density n, and hole density p.
Poisson equation
div(xe2x88x92xcex5grad("psgr"))=q(pxe2x88x92n+Ndxe2x88x92Na)xe2x80x83xe2x80x83(1)
Equation of continuity of electrons
div(Jn)=q(Rxe2x88x92G)xe2x80x83xe2x80x83(2)
Equation of continuity of holes
div(Jp)=xe2x88x92q(Rxe2x88x92G)xe2x80x83xe2x80x83(3)
Electron transport equation
Jn=xe2x88x92q*(xcexcn*n*grad("psgr")xe2x88x92Dn*grad(n))xe2x80x83xe2x80x83(4)
Hole transport equation
Jp=xe2x88x92q*(xcexcp*p*grad("psgr")+Dp*grad(p))xe2x80x83xe2x80x83(5)
where xcex5: permittivity of the medium, "psgr": electric potential (potential), Nd: donor density, Na: acceptor density, R: carrier re-coupling amount, G: carrier generation amount, n: electron density, p: hole density, xcexcn: electron mobility, xcexcp: hole mobility, Dn: electron diffusion coefficient, Dp: hole diffusion coefficient, Jn: electron current density, Jp: hole current density, q: unit electric charge.
Moreover, the electron diffusion coefficient Dn and the hole diffusion coefficient Dp satisfy the following equations, respectively.
Dn=xcexcn*k*T/q
Dp=xcexcp*k*T/q
where k: Boltzmann constant, and T: temperature.
According to the conventional simulation method, a given MOS type semiconductor device is divided into many micro regions. Moreover, mesh points (or grid points) are placed on each of the micro regions. Next, the electric potential, a electron density, and hole density at each of the mesh points are calculated. It is interpreted that these electric potential, electron density, and hole density at each of the mesh points represent the electric potential, electron density, and hole density in the micro region. By connecting the electric potentials, electron densities, and hole densities in all the micro regions, the distributions of electric potential, electron density, and hole density throughout the MOS type semiconductor device are calculated.
Next, a MOSFET structure simulator according to the prior art will be explained.
This MOSFET structure simulator has an input means to which simulation conditions such as the shape of the semiconductor device, the distribution of impurities, temperature, input voltage, and the like are input, a characteristic calculation means which calculates the device characteristics of the semiconductor device by solving prescribed equations of device physics based on the distribution information provided from the input setting means, and a judging means which judges whether the characteristics value calculated by the characteristic calculation means has converged or not. In some cases, an output means which outputs information containing the calculation result produced by the characteristic calculation means in a prescribed form is installed in the apparatus, and the entire configuration including the output means is called a simulator. The expression xe2x80x9cthe characteristics value has convergedxe2x80x9d refers to a state in which the amount of change of the characteristic value that changes in accordance with the simulation process has reached below a prescribed level.
The following calculations are carried out in the above-described characteristics calculation means.
(1) Discretization of the semiconductor device into mesh points
(2) Setting the initial values of the electric potential "psgr", electron density n, and hole density p
(3) Calculation of the vertical electric field Ev of the semiconductor device from the initial values of the electric potential "psgr", electron density n, and hole density p
(4) Calculation of the carrier mobility xcexc from the impurity concentration N, temperature T of the semiconductor device, and vertical electric field Ev
(5) Calculation of the electric potential "psgr", electron density n, and hole density p at each mesh point from the vertical electric field Ev and the carrier mobility xcexc
(6) Calculation of the electron current density Jn and hole current density Jp from the electric potential "psgr", electron density n, hole density p, and carrier mobility xcexc at each mesh point
(7) Calculation of the value of the electric current from the electron current density Jn and hole current density Jp
Next, the simulation method with the above-described MOSFET structure simulator will be explained with reference to the flow chart shown in FIG. 8.
First, conditions required for the simulation such as the shape of the semiconductor, the distribution of impurities, temperature, and the like are input to the input means (step S501).
Next, prescribed calculations are carried out by the characteristic calculation means. First, the semiconductor device is divided into meshes (step S502). Next, the initial values of the electric potential "psgr", electron density n, and hole density p are set based on the conditions defined in the simulation condition input process (step S501) (step S503). Using these values, the vertical electric field Ev of the semiconductor device is calculated (step S504).
Next, the carrier mobility xcexc is calculated from the impurity concentration N, temperature T of the semiconductor device, and vertical electric field Ev obtained in the previous process (step S505). The electric potential "psgr", electron density n, and hole density p at each mesh point are then calculated from the vertical electric field Ev and the carrier mobility xcexc obtained in the previous process (step S506). Next, the electron current density Jn and hole current density Jp are calculated at each mesh point from the electric potential "psgr", electron density n, hole density p, and carrier mobility xcexc using equations (4) and (5) (step S507). Moreover, the value of the electric current that flows through each terminal is obtained from the electron current density in and hole current density Jp calculated in the previous process (step S508).
Next, the judging means judges whether the value of the electric current calculated in the previous process by the characteristic calculation means has converged or not. First, it is judged whether the difference between the amount of the incoming current and the amount of the outgoing current lies in a prescribed convergence condition range or not (step S509). If the difference does not lie in the prescribed convergence condition range, the characteristic value calculation is repeated re-starting from the vertical electric field calculation process (step S504) using the electric potential "psgr", electron density n, and hole density p as the initial values obtained in the electric potential "psgr", electron density n, and hole density p calculation process (step S506). If the difference lies in the prescribed convergence condition range, the device characteristic of the semiconductor device is evaluated using the electric current value.
The simulation information and the characteristic value of the semiconductor device obtained in the previous processes are output and displayed by an output device inside the simulator or another output device.
It should be noted that, in a MOS type semiconductor device, the characteristic value in the neighborhood of the interface between the insulating material of the MOS semiconductor device and the semiconductor (hereafter, this will be referred to as xe2x80x9cinsulating material/semiconductor interfacexe2x80x9d) significantly influences the characteristic of the semiconductor device. Therefore, the carrier mobility xcexc on the insulating material/semiconductor interface needs to be calculated precisely.
Here, in the carrier mobility xcexc calculation process (step S505), the carrier mobility xcexc is defined as a function of the impurity concentration N, temperature T of the semiconductor device, and vertical electric field Ev. Many forms of this function are known. According to the prior art, the carrier mobility xcexc is calculated from an empirically derived equation. However, in the experiment, the carrier mobility xcexc at each mesh point of the semiconductor device cannot be measured. Instead, only the average carrier mobility in the inversion layer formed in the neighborhood of the insulating material/semiconductor interface can be measured. Therefore, a method for calculating the carrier mobility xcexc at each mesh point based on the experimentally obtained average mobility is needed. As an exemplary such method, as disclosed in the Japanese Patent Laid Open No. H10-4191, the method that uses the following Shin""s equation is known.
xcexc(Ev)=xcexceff(Eeff)+(Evxe2x88x92E0)dxcexceff(Eeff)/dEvxe2x80x83xe2x80x83(6)
where, xcexceff: the average mobility in the inversion layer formed in the insulating material/semiconductor interface, Eeff=xcex7Ev+(1xe2x88x92xcex7)E0 (effective vertical electric field), E0: vertical electric field outside the inversion layer, Ev: vertical electric field at mesh pint, xcex7=xc2xd (in the case of electrons), xcex7=⅓ (in the case of holes).
The method disclosed in the reference C. Lombmdi et al., xe2x80x9cA Physically Based Mobility Model For Numerical Simulation of Non Planer Devicesxe2x80x9d. IEEE Trans. CAD., vol. 7, 1164-1171, 1988 is another method for calculating the carrier mobility at each mesh point based on experimentally obtained average carrier mobility. In this reference, the carrier mobility xcexc at each mesh point is given by the following equation (7).
1/xcexc(Ev, N, T)=1/(b/Ev+c1*Nc2(T*Ev))+(Ev)2/d+1/xcexc2(N, T)xe2x80x83xe2x80x83(7)
where, xcexc2(N, T) is a prescribed function that depends on N and T, and
b=3.1*108,
c1=3.0*107,
c2=13, and
d=6*1014.
However, when the carrier mobility xcexc is calculated using these parameters, the obtained carrier mobility xcexc can deviate significantly from the actual carrier mobility. Hence, the measured value is reproduced using a matching means which judges whether the calculated electric current value agrees with the measured electric current value. In this way, the device characteristic can be simulated with a higher degree of accuracy. A simulation in which a matching judging means is used will be explained with reference to the flow chart shown in FIG. 3. The matching means has an agreement judging means which judges whether the electric current value calculated by the simulator agrees with the measured electric current value or not. Using this matching means, agreement judging process (step S610) which judges whether the electric current value calculated by the simulator agrees with the measured electric current value or not is performed after the simulator convergence judging process (step S609) is completed as shown in FIG. 9.
If it is judged that the electric current value calculated by the simulator does not agree with the measured electric current value obtained in the agreement judging process (step S610), parameter change process (step S611) for changing the parameters for the carrier mobility xcexc calculation equation in the carrier mobility xcexc calculation process (step S605) is carried out. For example, when equation (7) is used for the carrier mobility xcexc calculation equation, the parameters b, c1, c2, and d are changed. The parameter change process may be performed by the characteristic calculation means or matching means. After the parameter change process (step S611) is completed, the processes starting from the simulation condition input process (step S601) are repeated.
The above-described simulator and simulation in which the matching means is used has the following problems. FIG. 10 shows the current-voltage characteristic of the simulation result in which values measured from an N-type MOSFET (hereafter, this will be referred to as NMOS) and a matching means were used. The vertical axis represents the drain current Id, and the horizontal axis represents the gate voltage Vg. The matching means is characterized in that it reproduces the measured current values. The measured current values are reproduced satisfactorily when the voltage Vb applied to the back surface of the substrate (hereafter, this voltage will be referred to as substrate voltage) is 0. However, when the substrate voltage Vb is xe2x88x923V, the simulation result deviates significantly from the measured values.
FIG. 11 shows the current ratio and the carrier ratio in the case the substrate voltage Vb is 0V and in the case the substrate voltage Vb is xe2x88x923V. As shown in FIG. 11, the ratio between the simulated carrier amount in the case the substrate voltage Vb is 0V and the simulated carrier amount in the case the substrate voltage Vb is xe2x88x923V agrees approximately with the ratio between the measured carrier amount in the case the substrate voltage Vb is 0V and the measured carrier amount in the case the substrate voltage Vb is xe2x88x923V regardless of the gate voltage Vg.
FIG. 12 shows the distribution of the carrier mobility xcexc with respect to the distance X from the insulation material/semiconductor interface. FIG. 13 shows the distribution of the vertical electric field Ev with respect to the distance X from the insulation material/semiconductor interface. As shown in FIG. 12, on the insulation material/semiconductor interface, the carrier mobility xcexc at Vb=0V differs very little from the carrier mobility xcexc at Vb=xe2x88x923V. However, off the insulation material/semiconductor interface, the carrier mobility xcexc at Vb 0V differs significantly from the carrier mobility xcexc at Vb=xe2x88x923V. This phenomenon is caused by the difference between the distribution of the vertical electric field Ev in the neighborhood of the insulation material/semiconductor interface when Vb=0V and the distribution of the vertical electric field Ev in the neighborhood of the insulation material/semiconductor interface when Vb=xe2x88x923V as shown in FIG. 13.
However, no parameter that can correct the vertical electric field Ev while the mobility is calculated or depends on the substrate voltage exists. Hence, according to the prior art, it is difficult to achieve simultaneous fitting of the electric current characteristic that depends on the substrate voltage.
Given these problems, it is an object of the present invention to provide a simulation method and a simulator capable of reproducing the electric current characteristic that depends on the substrate voltage easily and accurately, solving these problems.